Course on Algebra and Coalgebra

This is a back up reference of the page of the course on Algebra and Coalgebra (Autumn 2013).

Modal Logic, Algebra and Coalgebra (Autumn 2013)

Contents of these pages

• Course Description and Prerequisites
• Practicalities

News

• For the exam you need to study the material that you can find here (that is, the indicated sections of the Algebra|Coalgebra chapter, together with the material provided for the last three classes on coalgebra).
• The deadline for submitting the sixth homework has been extended to January 6, 2014.
• An example of a coinductive proof can be found here; the slides of the full lecture (in Dutch, for teachers and high school kids) are here.

Practicalities

Staff

• Lecturers: Yde Venema (phone: 525 5299) and Luca Spada

Dates/location:

• Classes run from October 29 until December 11; there will be 14 classes in total.
• There are two classes weekly:
  • on Tuesdays from 09.00 – 10.45 in room A1.14, and
  • on Wednesdays from 13.00 – 14.45 in room G3.13

  Both rooms are in Science Park.

• Tuesday classes will be on Coalgebra, taught by Yde Venema, Wednesday classes will be on Algebra, taught by Luca Spada.

Course material

• The basic course material for the course is the following text:
  Y Venema, Algebras and Coalgebras,
  in: J van Benthem, P Blackburn and F Wolter (editors), Handbook of Modal Logic,
  Elsevier, Amsterdam, 2006, pp 331-426.

Grading

• Grading is through homework assignme and a final exam.

Course Description

Modal languages are simple yet expressive and flexible tools for describing all kinds of relational structures. Thus modal logic finds applications in many disciplines such as computer science, mathematics, linguistics or economics. Notwithstanding this enormous diversity in appearance and application area, modal logics have a great number of properties in common. This common mathematical backbone forms the topic of this course. This year the course will focus on algebraic and coalgebraic aspects of modal logic.

Content

More specifically, we will cover (at least) the following topics:

• Algebra: Boolean algebras, modal algebras, boolean algebras with operators, algebraizing modal logic, Lindenbaum-Tarski algebras, free algebras, complex/discrete duality, topologial duality, varieties of BAOs and their properties, canonicity.
• Coalgebra: set functors and their coalgebras, final coalgebra, bisimilarity and behavioural equivalence, coinduction and covarieties, coalgebraic modal logic (both based on relation lifting and on predicate liftings), algebra and coalgebra.

Prerequisites

It is assumed that students entering this class possess

• a working knowledge of modal logic (roughly corresponding to the first sections of the Chapters 1-4 of the Modal Logic book by Blackburn, de Rijke and Venema).
• some mathematical maturity.

Basic knowledge of general algebra, topology and category theory will be handy but not necessary.

Course on Capita Selecta in Modal Logic

Starting from the 29th of October I will start teaching a course with Yde Venema at the University of Amsterdam on Algebra and Coalgebra.  The webpage of the course can be found here.

Matematica Discreta e Logica Matematica (Informatica)

This year I will give a series of introductory lectures on Mathematical Logic for the course “Matematica Discreta e Logica Matematica” at the B.Sc. in Computer Science.  I will try to write down some lecture notes, but in the meanwhile students may find helpful to consult some references.  Probably any book on mathematical logic contains the topics I am about to discuss, yet here there are some suggestions aimed at Italian speakers without a strong background in mathematics.

1. Giangiacomo Gerla Linguaggio e verità. ilmiolibro.it. 2011. 274 pages (also available in reduced form, for free, here)
2. Andrea Asperti, Agata Ciabattoni Logica a informatica. McGraw-Hill. 1997. 204 pages.
3. Dario Palladino Corso di logica. Introduzione al calcolo dei predicati. Carocci. 2010. 409 pages.

Tutorial on Lukasiewicz logic and MV-algebras at the summer school in Campinas.

Prior to the MaToMUVI conference Philosophy and Mathematics of Uncertainty and Vagueness organised at CLE (UniCamp, Brazil), there will be a summer school for young researchers and undergaduate students.  I will give a short tutorial on MV-algebras (seen in a broad perspective).  I (hand-wrote) some notes that can be downloaded here.  Take it cum grano salis, they are just notes I have used to follow a line of thoughts, still I believe they can be useful to the audience as they contain both specific references to the results presented and some good books giving an updated account of the subject.

Logica II per Informatica

This year I will teach the course “Logica II” for the M.sc. degree in Computer Science.  I think I will follow Moore’s modified method for the course (see also the wikipedia entry for the original method).

There are some freely available notes, looking promising,  I will use during the course.

Other reference (standard books) that may be used as sources of inspiration during the course are:

• Mendelson, E.  Introduction to Mathematical Logic. Chapman & Hall 2009.
• Mundici, D.  Logica – Metodo breve. Springer Verlag 2011.
• Asperti, A. and Ciabattoni, A. Logica a Informatica. McGraw-Hill 2003

Further information will follow on this website.